You'll get a notification in your Inbox when there are updates to tasks you care about / are subscribed to. Specifically, if you get assigned to a task, @mentioned on a task, or someone replies to your messages, you will be notified when there are updates.
Notifications to your Inbox are real-time — so as soon as there's an update, your Inbox will be badged.
Notifications will stay in your Inbox until you archive them (
CMD+E ), but you can always see archived conversations from the
... menu if you want to find one you've already archived.